Smoothness and Classicality on eigenvarieties

TL;DR

This paper proves new cases of a conjecture linking classicality of p-adic automorphic forms to their Galois representations using patched eigenvarieties, advancing understanding in automorphic forms and Galois representations.

Contribution

It establishes new classicality results for overconvergent automorphic forms in higher dimensions via patched eigenvarieties, extending previous conjectures.

Findings

Proved classicality in new cases for automorphic forms with crystalline Galois representations.

Utilized patched eigenvarieties to analyze eigenvariety geometry.

Extended classicality results to arbitrary dimension.

Abstract

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimension by making crucial use of the "patched eigenvariety".